Reaction-diffusion with memory in the minimal state framework
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- by Monica Conti, Elsa M. Marchini and Vittorino Pata PDF
- Trans. Amer. Math. Soc. 366 (2014), 4969-4986 Request permission
Abstract:
We consider the integrodifferential equation \[ \partial _t u-\Delta u -\int _0^\infty \kappa (s)\Delta u(t-s) \mathrm {d} s + \varphi (u)=f\] arising in the Coleman-Gurtin theory of heat conduction with hereditary memory. Within a novel abstract framework, based on the notion of minimal state, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related semigroup of solutions.References
- A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492
- Mickaël D. Chekroun, Francesco Di Plinio, Nathan E. Glatt-Holtz, and Vittorino Pata, Asymptotics of the Coleman-Gurtin model, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), no. 2, 351–369. MR 2746378, DOI 10.3934/dcdss.2011.4.351
- V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46 (2006), no. 3-4, 251–273. MR 2215885
- Vladimir V. Chepyzhov and Mark I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1868930, DOI 10.1051/cocv:2002056
- Bernard D. Coleman and Morton E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967), 199–208 (English, with German summary). MR 214334, DOI 10.1007/BF01596912
- Monica Conti and Elsa M. Marchini, Wave equations with memory: the minimal state approach, J. Math. Anal. Appl. 384 (2011), no. 2, 607–625. MR 2825211, DOI 10.1016/j.jmaa.2011.06.009
- Monica Conti, Elsa M. Marchini, and Vittorino Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst. 27 (2010), no. 4, 1535–1552. MR 2629536, DOI 10.3934/dcds.2010.27.1535
- Monica Conti, Vittorino Pata, and Marco Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J. 55 (2006), no. 1, 169–215. MR 2207550, DOI 10.1512/iumj.2006.55.2661
- Constantine M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297–308. MR 281400, DOI 10.1007/BF00251609
- Gianpietro Del Piero and Luca Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138 (1997), no. 1, 1–35. MR 1463802, DOI 10.1007/s002050050035
- Luca Deseri, Mauro Fabrizio, and Murrough Golden, The concept of minimal state in viscoelasticity: new free energies and applications to PDEs, Arch. Ration. Mech. Anal. 181 (2006), no. 1, 43–96. MR 2221203, DOI 10.1007/s00205-005-0406-1
- Claudio Giorgi, Vittorino Pata, and Alfredo Marzocchi, Uniform attractors for a non-autonomous semilinear heat equation with memory, Quart. Appl. Math. 58 (2000), no. 4, 661–683. MR 1788423, DOI 10.1090/qam/1788423
- H. Grabmüller, On linear theory of heat conduction in materials with memory. Existence and uniqueness theorems for the final value problem, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 2, 119–137. MR 446112, DOI 10.1017/S0308210500013937
- Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
- Alain Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 17, Masson, Paris, 1991 (French). MR 1084372
- Mauro Fabrizio, Claudio Giorgi, and Vittorino Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal. 198 (2010), no. 1, 189–232. MR 2679371, DOI 10.1007/s00205-010-0300-3
- S.-O. Londen and J. A. Nohel, Nonlinear Volterra integro-differential equation occurring in heat flow, J. Integral Equations 6 (1984), no. 1, 11–50. MR 727934
- R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), no. 2, 313–332. MR 515894, DOI 10.1016/0022-247X(78)90234-2
- A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: evolutionary equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 103–200. MR 2508165, DOI 10.1016/S1874-5717(08)00003-0
- Jace W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204. MR 295683, DOI 10.1090/S0033-569X-1971-0295683-6
- Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967, DOI 10.1007/978-1-4684-0313-8
Additional Information
- Monica Conti
- Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
- Email: monica.conti@polimi.it
- Elsa M. Marchini
- Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
- Email: elsa.marchini@polimi.it
- Vittorino Pata
- Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
- MR Author ID: 358540
- Email: vittorino.pata@polimi.it
- Received by editor(s): October 17, 2011
- Received by editor(s) in revised form: February 1, 2013
- Published electronically: November 6, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 4969-4986
- MSC (2010): Primary 35B41, 35K05, 45K05, 47H20
- DOI: https://doi.org/10.1090/S0002-9947-2013-06097-7
- MathSciNet review: 3217706